Mathematics > Category Theory

Title:
The inner automorphism 3-group of a strict 2-group

Abstract: Any group $G$ gives rise to a 2-group of inner automorphisms,
$\mathrm{INN}(G)$. It is an old result by Segal that the nerve of this is the
universal $G$-bundle. We discuss that, similarly, for every 2-group $G_{(2)}$
there is a 3-group $\mathrm{INN}(G_{(2)})$ and a slightly smaller 3-group
$\mathrm{INN}_0(G_{(2)})$ of inner automorphisms. We describe these for
$G_{(2)}$ any strict 2-group, discuss how $\mathrm{INN}_0(G_{(2)})$ can be
understood as arising from the mapping cone of the identity on $G_{(2)}$ and
show that its underlying 2-groupoid structure fits into a short exact sequence
$G_{(2)} \to \mathrm{INN}_0(G_{(2)}) \to \Sigma G_{(2)}$.
As a consequence, $\mathrm{INN}_0(G_{(2)})$ encodes the properties of the
universal $G_{(2)}$ 2-bundle.

Comments:

references added, relation to simplicial constructions expanded, version to appear in JHRS